Optimal. Leaf size=82 \[ \frac{x^2 \sqrt [4]{a+b x^4}}{3 b}-\frac{2 a^{3/2} \left (\frac{b x^4}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{3 b^{3/2} \left (a+b x^4\right )^{3/4}} \]
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Rubi [A] time = 0.045413, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {275, 321, 233, 231} \[ \frac{x^2 \sqrt [4]{a+b x^4}}{3 b}-\frac{2 a^{3/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 b^{3/2} \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 275
Rule 321
Rule 233
Rule 231
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^4\right )^{3/4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=\frac{x^2 \sqrt [4]{a+b x^4}}{3 b}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )}{3 b}\\ &=\frac{x^2 \sqrt [4]{a+b x^4}}{3 b}-\frac{\left (a \left (1+\frac{b x^4}{a}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{3 b \left (a+b x^4\right )^{3/4}}\\ &=\frac{x^2 \sqrt [4]{a+b x^4}}{3 b}-\frac{2 a^{3/2} \left (1+\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 b^{3/2} \left (a+b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.020044, size = 64, normalized size = 0.78 \[ \frac{x^2 \left (-a \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )+a+b x^4\right )}{3 b \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{{x}^{5} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{5}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.1024, size = 27, normalized size = 0.33 \begin{align*} \frac{x^{6}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6 a^{\frac{3}{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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